House of Math

What Is the Median of Grouped Data?

Here you’ll learn how to find the median of grouped data. In this case, you find the median by reading off a graph or through calculation. Here’s the procedure for both methods.

Reading off a Graph

To find the median in a set of grouped data, you need to draw the graph representing the relative cumulative frequency. This is how you do it:

Rule

How to Draw the Graph Representing the Relative Cumulative Frequency

1.: If you don’t have a table containing the intervals, you have to group your data into intervals. Unless specified by the exercise, you can choose the sizes of the intervals yourself. Make sure to choose intervals in such a way that they reveal the information you’re looking for. You’ll know your intervals are good if you can say “yes” when you ask yourself “are my intervals reasonable?”
2.: As you’re determining the relative cumulative frequency, your $y$ values will be decimal numbers between 0 and 1, or a percentage between $0$ % and $100$ %.
3.: The first point of the graph is decided in a different way than the rest of the points! Its $x$ value is equal to the lowest value in the first interval and its $y$ value is $0$ %.
4.: Then, mark all the points in a coordinate system and draw lines between the points.

For grouped data, you can assume the values are evenly distributed inside each interval. That’s why you can find the median of grouped data by reading off the graph when the relative cumulative frequency is known.

Rule

Median of Grouped Data: Graphical

You find the median where the relative cumulative frequency passes $0.5$ , or $50$ %.

Through Calculation

When you need to find the median through calculation, you have to find the expression for a line between two points. These points are taken from the interval where the relative cumulative frequency passes $50$ %, and the interval before it. Here’s how to do it:

Rule

Median of Grouped Data: Calculation

1.: To find the expression for the line, $y = a x + b$ , you have to decide which points to use to find $a$ and $b$ .
2.: For the $x$ values, use the smallest and greatest values of the interval that passes $50$ % or $0.5$ . You can call these values $x_{1}$ and $x_{2}$ respectively.
3.: Use the relative cumulative frequency of the interval that passes $50$ % as $y_{2}$ and the relative cumulative frequency of the interval before as $y_{1}$ .
4.: You’ve now found two points: $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
5.: Calculate $a$ using the formula for the slope of a straight line: $a = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .
6.: Calculate $b$ by using the formula $y = a x + b$ . Insert the value you found for $a$ and the $x$ and $y$ coordinates of a point on the line (it’s smart to use one of the points we found above). Solve the equation for $b$ .
7.: Insert the values you’ve found for $a$ and $b$ into the expression for the straight line $y = a x + b$ .
8.: Set $y = a x + b = 0.5$ for a decimal number or $50$ % for a percentage, and solve for $x$ .
9.: This value of $x$ is your median.

Example 1

Below you can see a table showing the relative cumulative frequency of the weight of newborn children.


Weight	Relative Cumulative Frequency

$[0 kg, 2.5 kg)$	$1.2$ %

$[2.5 kg, 3.0 kg)$	$9.4$ %

$[3.0 kg, 3.5 kg)$	$42.2$ %

$[3.5 kg, 4.0 kg)$	$78$ %

$[4.0 kg, 4.5 kg)$	$96$ %

$[4.5 kg, 5.0 kg)$	$99.6$ %

$[5.0 kg, 5.5 kg)$	$100$ %

From the table, you can see that the relative cumulative frequency passes $50$ % in the interval $[3.5 kg, 4.0 kg)$ . That means you have $x_{1} = 3.5$ , $x_{2} = 4.0$ , $y_{1} = 42.2$ and $y_{2} = 78$ . That gives us the points $(3.5, 42.2)$ and $(4.0, 78)$ . Calculate $a$ by using these values in the formula

a = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{78 - 42.2}{4.0 - 3.5} = 71.6

Calculate $b$ by inserting the point $(3.5, 42.2)$ and the value of $a$ into $y = a x + b$ : $\begin{array}{l} 42.2 & = 71.6 \cdot 3.5 + b \\ = 250.6 + b \\ b & = - 208.4 \end{array}$

You then get the expression for the line:

y = 71.6 x - 208.4

Set it equal to $50$ (as our numbers are percentages) and solve for $x$ : $\begin{array}{l} 71.6 x - 208.4 & = 50 \\ 71.6 x & = 258.9 \\ x & = 3.61 kg \end{array}$